Cyclic coverings of genus 2 curves of Sophie Germain type

We consider cyclic unramified coverings of degree $d$ of irreducible complex smooth genus 2 curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order $d$. The rich geometry of the associated Prym map has been studied in se...

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Autores: Naranjo del Val, Juan Carlos, Ortega Ortega, Angela, Spelta, Irene
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2024
País:España
Institución:Universidad de Barcelona
Repositorio:Dipòsit Digital de la UB
OAI Identifier:oai:diposit.ub.edu:2445/222606
Acceso en línea:https://hdl.handle.net/2445/222606
Access Level:acceso abierto
Palabra clave:Formes de Jacobi
Varietats abelianes
Corbes algebraiques
Jacobi forms
Abelian varieties
Algebraic curves
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spelling Cyclic coverings of genus 2 curves of Sophie Germain typeNaranjo del Val, Juan CarlosOrtega Ortega, AngelaSpelta, IreneFormes de JacobiVarietats abelianesCorbes algebraiquesJacobi formsAbelian varietiesAlgebraic curvesWe consider cyclic unramified coverings of degree $d$ of irreducible complex smooth genus 2 curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order $d$. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2,3,5,7$ are quite well understood. Nevertheless, very little is known for higher values of $d$. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d \geq 11$ prime such that $\frac{d-1}{2}$ is also prime. We use results of arithmetic nature on $G L_2$-type abelian varieties combined with theta-duality techniques.2024info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://hdl.handle.net/2445/222606Articles publicats en revistes (Matemàtiques i Informàtica)reponame:Dipòsit Digital de la UBinstname:Universidad de BarcelonaInglésReproducció del document publicat a: https://doi.org/doi:10.1017/fms.2024.422024, vol. 12https://doi.org/doi:10.1017/fms.2024.42cc-by (c) J.C. Naranjo et al., 2024http://creativecommons.org/licenses/by/3.0/es/info:eu-repo/semantics/openAccessoai:diposit.ub.edu:2445/2226062026-05-27T06:46:51Z
dc.title.none.fl_str_mv Cyclic coverings of genus 2 curves of Sophie Germain type
title Cyclic coverings of genus 2 curves of Sophie Germain type
spellingShingle Cyclic coverings of genus 2 curves of Sophie Germain type
Naranjo del Val, Juan Carlos
Formes de Jacobi
Varietats abelianes
Corbes algebraiques
Jacobi forms
Abelian varieties
Algebraic curves
title_short Cyclic coverings of genus 2 curves of Sophie Germain type
title_full Cyclic coverings of genus 2 curves of Sophie Germain type
title_fullStr Cyclic coverings of genus 2 curves of Sophie Germain type
title_full_unstemmed Cyclic coverings of genus 2 curves of Sophie Germain type
title_sort Cyclic coverings of genus 2 curves of Sophie Germain type
dc.creator.none.fl_str_mv Naranjo del Val, Juan Carlos
Ortega Ortega, Angela
Spelta, Irene
author Naranjo del Val, Juan Carlos
author_facet Naranjo del Val, Juan Carlos
Ortega Ortega, Angela
Spelta, Irene
author_role author
author2 Ortega Ortega, Angela
Spelta, Irene
author2_role author
author
dc.subject.none.fl_str_mv Formes de Jacobi
Varietats abelianes
Corbes algebraiques
Jacobi forms
Abelian varieties
Algebraic curves
topic Formes de Jacobi
Varietats abelianes
Corbes algebraiques
Jacobi forms
Abelian varieties
Algebraic curves
description We consider cyclic unramified coverings of degree $d$ of irreducible complex smooth genus 2 curves and their corresponding Prym varieties. They provide natural examples of polarized abelian varieties with automorphisms of order $d$. The rich geometry of the associated Prym map has been studied in several papers, and the cases $d=2,3,5,7$ are quite well understood. Nevertheless, very little is known for higher values of $d$. In this paper, we investigate whether the covering can be reconstructed from its Prym variety, that is, whether the generic Prym Torelli theorem holds for these coverings. We prove this is so for the so-called Sophie Germain prime numbers, that is, for $d \geq 11$ prime such that $\frac{d-1}{2}$ is also prime. We use results of arithmetic nature on $G L_2$-type abelian varieties combined with theta-duality techniques.
publishDate 2024
dc.date.none.fl_str_mv 2024
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/2445/222606
url https://hdl.handle.net/2445/222606
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Reproducció del document publicat a: https://doi.org/doi:10.1017/fms.2024.42
2024, vol. 12
https://doi.org/doi:10.1017/fms.2024.42
dc.rights.none.fl_str_mv cc-by (c) J.C. Naranjo et al., 2024
http://creativecommons.org/licenses/by/3.0/es/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv cc-by (c) J.C. Naranjo et al., 2024
http://creativecommons.org/licenses/by/3.0/es/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv Articles publicats en revistes (Matemàtiques i Informàtica)
reponame:Dipòsit Digital de la UB
instname:Universidad de Barcelona
instname_str Universidad de Barcelona
reponame_str Dipòsit Digital de la UB
collection Dipòsit Digital de la UB
repository.name.fl_str_mv
repository.mail.fl_str_mv
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